Inertially arbitrary patterns

An n×n sign pattern matrix A is an inertially arbitrary pattern (IAP) if each non-negative triple (rst) with r+s+t=n is the inertia of a matrix with sign pattern A. This paper considers the n×n(n≥2) skew-symmetric sign pattern S_n with each upper off-diagonal entry positive, the (1,1) entry negative, the (nn) entry positive, and every other diagonal entry zero. We prove that S_n is an IAP.     

An extremal problem on potentially Kr,s-graphic sequences

We consider a variation of a classical Turán-type extremal problem (F. Chung, R. Graham, Erd s on Graphs: His Legacy of Unsolved Problems, AK Peters Ltd., Wellesley, 1998, Chapter 3) as follows: Determine the smallest even integer σ(K_r,s,n) such that every n-term graphic sequence π=(d₁,d₂,…,d_n) with term sum σ(π)=d₁+d₂++d_nσ(K_r,s,n) is potentially K_r,s-graphic, where K_r,s is a r×s complete bipartite graph, i.e., π has a realization G containing K_r,s as its subgraph. In this paper, we first give sufficient conditions for a graphic sequence being potentially K_r,s-graphic, and then we determine σ(K_r,r,n) for r=3,4.     

Maximal IM-unextendable graphs

A graph G is maximal IM-unextendable if G is not induced matching extendable and, for every two nonadjacent vertices x and y, G+xy is induced matching extendable. We show in this paper that a graph G is maximal IM-unextendable if and only if G is isomorphic to M_r(K_s(K_n1K_n2K_nt)), where M_r is an induced matching of size r, r1, t=s+2, and each n_i is odd.     

A matrix problem associated with computing Cohen-Macaulay type

Suppose A∈M_n×m(F), B∈M_n×t(F) for some field F. Define Г(AB) to be the set of n×n diagonal matrices D such that the column space of DA is contained in the column space of B. In this paper we determine dim Г(AB). For matrices AB of the same rank we provide an algorithm for computing dim Г(AB).     

Bondage number of strong product of two paths

The bondage number b(G) of a graph G is the cardinality of a minimum set of edges whose removal from G results in a graph with a domination number greater than that of G. In this paper, we obtain the exact value of the bondage number of the strong product of two paths. That is, for any two positive integers m≥2 and n≥2, b(P_m?P_n) = 7 - r(m) - r(n) if (r(m), r(n)) = (1, 1) or (3, 3), 6 - r(m) - r(n) otherwise, where r(t) is a function of positive integer t, defined as r(t) = 1 if t ≡ 1 (mod 3), r(t) = 2 if t ≡ 2 (mod 3), and r(t) = 3 if t ≡ 0 (mod 3).     

Manifolds of idemopotent matrices

Let M_n be the set of n×n matrices and r a nonnegative integer with r ≤ n. It is known,from Lie groups, that the rank r idempotent matrices in M_n form an arcwise connected 2n (n-r)-dimensional analytic manifold. This paper provides an elementary proof of this result making it accessible to a larger audience.     

Multiplicative properties of generalized matrix functions

We consider scalar-valued matrix functions for n×n matrices A=(a_ij) defined by Where G is a subgroup of S_n the group of permutations on n letters, and χ is a linear character of G. Two such functions are the permanent and the determinant. A function (1) is multiplicative on a semigroup S of n×n matrices if d(AB)=d(A)d(B) AB∈S.

With mild restrictions on the underlying scalar ring we show that every element of a semigroup containing the diagonal matrices on which (1) is multiplicative can have at most one nonzero diagonal(i.e., diagonal with all nonzero entries)and conversely, provided that χ is the principal character(χ≡1).     

Caractérisation spectrale de la forme de Jordan

Soit , où désigne l'ensemble des matrices n×n à coefficients complexes. Nous montrons qu'on peut complètement caractériser la forme de Jordan de A en examinant le polynôme caractéristique de tA+X pour tous les tC et tous les . Ceci nous permet de donner une démonstration plus élémentaire d'un théorème de Baribeau et Ransford sur les transformations holomorphes de qui préservent le spectre.

Denote by the set of complex n×n matrices, and let . We give a variational, purely spectral characterization of the Jordan form of A by examining the characteristic polynomial of the perturbed matrices tA+X for tC and . This allows us to give a more elementary proof of a theorem of Baribeau and Ransford on spectrum-preserving holomorphic maps on .     

Linear preservers of matrices of rank-2

Let T be a linear operator on the space of all m×n matrices over any field. we prove that if T maps rank-2 matrices to rank-2 matrices then there exist nonsingular matrices U and V such that either T(X)=UXV for all matrices X, or m=n and T(X)=UX^tV for all matrices X where X^t denotes the transpose of X.     

带松弛条件的图的强边着色

给定两个非负整数s和t,图G的(s,t)-松弛强k边着色可表示为映射c:E(G)→[k],这个映射满足对G中的任意一条边e,颜色c(e)在e的1-邻域中最多出现s次并且在e的2-邻域中最多出现t次.图G的(s,t)-松弛强边着色指数,记作x'(s,t)(G),表示使得图G有(s,t)-松弛强k边着色的最小k值.在图G中...     

On the limit of a product of matrix exponentials

The existence of the limit as ε→0 exp[(A+B/ε)t] exp[-Bt/ε]is studied for n×n matrices A,B. Necessary and sufficient conditions on B that the limit exist for all A are given.     

Length-bounded disjoint paths in planar graphs

The following problem is considered: given: an undirected planar graph G=(V,E) embedded in , distinct pairs of vertices {r₁,s₁},…,{r_k,s_k} of G adjacent to the unbounded face, positive integers b₁,…,b_k and a function ; find: pairwise vertex-disjoint paths P₁,…,P_k such that for each i=1,…,k, P_i is a r_i−s_i-path and the sum of the l-length of all edges in P_i is at most b_i. It is shown that the problem is NP-hard in the strong sense. A pseudo-polynomial-time algorithm is given for the case of k=2.     

The chromatic difference sequence of the Cartesian product of graphs

The chromatic difference sequence cds(G) of a graph G with chromatic number n is defined by cds(G) = (a(1), a(2),…, a(n)) if the sum of a(1), a(2),…, a(t) is the maximum number of vertices in an induced t-colorable subgraph of G for t = 1, 2,…, n. The Cartesian product of two graphs G and H, denoted by G□H, has the vertex set V(G□H = V(G) x V(H) and its edge set is given by (x₁, y₁)(x₂, y₂) ε E(G□H) if either x₁ = x₂ and y₁ y₂ ε E(H) or y₁ = y₂ and x₁x₂ ε E(G).

We obtained four main results: the cds of the product of bipartite graphs, the cds of the product of graphs with cds being nondrop flat and first-drop flat, the non-increasing theorem for powers of graphs and cds of powers of circulant graphs.     

Ramsey linear families and generalized subdivided graphs

The following results are obtained. (i) Let p, d, and k be fixed positive integers, and let G be a graph whose vertex set can be partitioned into parts V₁, V₂,…, V_a such that for each i at most d vertices in V₁ … V_i have neighbors in V_i+1 and r(K_k, V_i) p | V(G) |, where V_i denotes the subgraph of G induced by V_i. Then there exists a number c depending only on p, d, and k such that r(K_k, G)c | V(G) |. (ii) Let d be a positive integer and let G be a graph in which there is an independent set I V(G) such that each component of G−I has at most d vertices and at most two neighbors in I. Then r(G,G)c | V(G) |, where c is a number depending only on d. As a special case, r(G, G) 6 | V(G) | for a graph G in which all vertices of degree at least three are independent. The constant 6 cannot be replaced by one less than 4.     

On the inertia of the Lyapunov transform AH+HA* for H >0

Given an arbitrary n×n matrix A with complex entries, we characterize all inertia triples (abc) that are attained by the Lyapunov transform AH+ HA^*, as H varies over the set of all n× n positive definite matrices.     

Chapter 4:algebraic sets, polynomials, and other functions

Let T be a linear operator on the vector space V ofn×n matrices over a field F. We discuss two types of problems in this chapter. First, what can we say about T if we assume that T maps a given algebraic set such as the special linear group into itself? Second, let p(x) be a polynomial function (such as det) on V into F. What can we say about T if Tpreserves p(x), i.e. p(T(X)) = p(X) for all X in V?     

The maximun of i(XAX-1+B)

Let F be an algebraically closed field. We denote by i(A) the number of invariant polynomials of a square matrix A, which are different from 1. For A,B any n×n matrices over F, we calculate the maximum of i(XAX^-1+B), where X runs over the set of all non-singular n×n matrices over F.     

Circular permutation graph family with applications

In a circular permutation diagram, there are two sets of terminals on two concentric circles: C_in and C_out. Given a permutation Π = [π₁, π₂, …, π_n], terminal i on C_in and terminal π_i on C_out are connected by a wire. The intersection graph G_c of a circular permutation diagram D_c is called a circular permutation graph of a permutation Π corresponding to the diagram D_c. The set of all circular permutation graphs of a permutation Π is called the circular permutation graph family of permutation Π. In this paper, we propose the following: (1) an O(V + E) time algorithm to check if a labeled graph G = (V, E) is a labeled circular permutation graph. (2) An O(n log n + nt) time algorithm to find a maximum independent set of a family, where n = Π and t is the cardinality of the output. (Number t in the worst case is O(n). However, if Π is uniformly distributed (and independent from i), its expected value is O(√n).) (3) An O(min(δV_clog logV_c,V_clogV_c) + E_c) time algorithm for finding a maximum independent set of a circular permutation diagram D_c, where δ is the minimum degree of vertices in the intersection graph G_c = (V_c,E_c) of D_c. (4) An O(n log log n) time algorithm for finding a maximum clique and the chromatic number of a circular permutation diagram, where n is the number of wires in the diagram.     

An improved product construction for large sets of Kirkman triple systems

It has been shown by Lei, in his recent paper, that there exists a large set of Kirkman triple systems of order uv (LKTS(uv)) if there exist an LKTS(v), a TKTS(v) and an LR(u), where a TKTS(v) is a transitive Kirkman triple system of order v, and an LR(u) is a new kind of design introduced by Lei. In this paper, we improve this product construction by removing the condition “there exists a TKTS(v)”. Our main idea is to use transitive resolvable idempotent symmetric quasigroups instead of TKTS. As an application, we can combine the known results on LKTS and LR-designs to obtain the existence of an LKTS(3ⁿm(2·13^n₁+1)(2·13^n_t+1)) for n1, m{1,5,11,17,25,35,43,67,91,123}{2^2r+125^s+1 : r0,s0}, t0 and n_i1 (i=1,…,t).     

Linear recurrence relations in the algebra of matrices and applications

We study here some linear recurrence relations in the algebra of square matrices. With the aid of the Cayley–Hamilton Theorem, we derive some explicit formulas for Aⁿ (nr) and e^tA for every r×r matrix A, in terms of the coefficients of its characteristic polynomial and matrices A^j, where 0jr−1.